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In order to motivate examples in the first class in congruence theory, my teacher remarked that the beginning chapters of the Holy Bible mathematically said entail the following: "Let the days of the week be congruent modulo seven."
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UnknownJun 12 '10 at 17:07

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Why did a question with so much positive feedback get closed?
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RomeoNov 28 '10 at 23:21

@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing.
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Qiaochu YuanNov 12 '13 at 3:03

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." --- John von Neumann. (From a 1947 ACM keynote, recalled by Alt in this 1972 CACM article.)

“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.” Lucien Szpiro during Algebra 1 lecture.

This reminds me of Surely You're Joking, Mr. Feynman!, in which the physics grad students at Princeton propose the theorem that mathematicians only prove trivial theorems.
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Pete L. ClarkJan 8 '10 at 6:27

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I once heard Henry McKean say of mathematical models in physics that "First, they’re pretty disgraceful. Second, they work extremely well...One of the faults of mathematicians is: when physicists give them an equation, they take it absolutely seriously." (I wrote this down on the spot.)
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Steve HuntsmanApr 23 '10 at 15:48

"The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps..."

Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can. --J. von Neumann's housekeeper, describing her employer.

The weatherman at Kitty Hawk thought Wilbur Wright wasted a lot of time watching gulls fly. And Darwin's housekeeper thought he would get more work done if he didn't keep staring at the ants in an anthill.
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SixWingedSeraphNov 30 '09 at 1:22

"The goal of this thesis is to furnish its author with the title of Doctor" is perhaps closer still. A. Douady was indeed a rather interesting person! [He was my co-supervisor, along with John Hubbard].
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Jacques CaretteMar 26 '10 at 2:30

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the sentence ends with : "and (some set) with (some structure)", playing on the two meanings of "munir" = furnish
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Feldmann DenisAug 24 '11 at 3:04

Obviously there is something wrong with ZF for proving their equivalence. I blame infinity.
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Andrew CritchNov 30 '09 at 1:36

4

Zorn's Lemma seems the most intuitive out of the three, but well-ordering isn't so counter-intuitive, since all it comes down to is being able to well-order a set with cardinality strictly larger than the natural numbers. Thinking about well-ordering the reals gives a false impression of the difficulty, since the well-ordering only has to do with the underlying set.
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Harry GindiNov 30 '09 at 6:25

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@Harry: I can't really agree with either statement. Naively, if we could truly "see" a well-ordering of a set of continuum cardinality, then intuitively we should be able to compare it to $\aleph_1$ and "see" whether it is larger.
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Todd Trimble♦Jul 3 '11 at 12:07

Along those lines, R. W. Hamming said "Mathematicians stand on each other’s shoulders while computer scientists stand on each other’s toes."
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John D. CookNov 30 '09 at 3:02

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My version of this quote, after working my way backwards through a series of papers, each relying on the previous ones: If I can't see a darn thing it's because I stand on the shoulders of giants...
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Ehud FriedgutDec 2 '09 at 5:49

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What was Gell-Mann's version? Something like, "If I have seen farther than others, it's because I'm surrounded by pygmies?"
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Todd Trimble♦Jan 31 '11 at 1:15

You might have gotten it from the Monthly in 1993, or a mailing list or blog that derives from that source. Google Books traces it back to a book about Ernst Cassirer published in 1949. Since it is quoted there as Cassirer's verbal story, it's less clear what Hilbert himself said. archive.org/stream/philosophyoferns033109mbp/…
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Greg KuperbergNov 29 '09 at 22:14

"Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate." - David Mumford

Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass:

"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance."

Grothendieck, of course, always pioneered this approach, and considered for example that Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way.

A friend who I won't name at the moment once told me this, paraphrased: An excellent problem-solver might not always be a great mathematician, while a bad problem-solver can still be an okay mathematician. On the other hand, a good Grothendieck is a great mathematician, while a bad Grothendieck is really terrible!
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Greg KuperbergDec 18 '09 at 7:20

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Greg, the symmetry in that statement would be nicer if you replaced "problem-solver" with "Erdos."
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Qiaochu YuanDec 27 '09 at 8:11

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That's a good suggestion, but I can't edit comments. So let me just rephrase the aphorism: "An excellent Erdos might reach certain limits as a mathematician, while a bad Erdos can still be an okay mathematician. On the other hand, a good Grothendieck can be a truly great mathematician, while a bad Grothendieck is really terrible."
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Greg KuperbergJan 19 '10 at 19:36

I don't know how much I agree with that. It may be true about physics, but not math.
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Harry GindiNov 30 '09 at 6:20

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I feel that it's true of both mathematics and physics, but when talking about mathematics it's a much deeper statement.
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Dan PiponiDec 2 '09 at 19:37

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I may have a surprise for you... "Understanding"="getting used to"! :)
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efqDec 4 '09 at 21:20

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I constantly try to check myself to make sure that I am really understanding, and am not just "getting used to" the things that I am learning. It is very difficult, but I think that I am gaining understanding of the basics little by little.
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Steven GubkinDec 7 '09 at 2:04

Here's a simple example. What is the radius of convergence for the power series of 1/(x^2 + 1) centered at 0? Looking only at the real line there's no apparent reason for the radius to be only 1. But in the complex plane, you can see that the radius is 1 because that's the distance from the center to the singularity at i. Another example would be using contour integration to compute integrals over the real line.
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John D. CookMar 26 '10 at 18:35

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon: Methods of modern mathematical physics)

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.

Bill, would you mind elaborating? As someone not particularly familiar with either field, I can imagine that by combinatorics being "discrete functional analysis" you mean e.g. generating function methods, or perhaps the general ambition of associating some sort of linear operator to combinatorial objects (e.g. adjacency matrix). But what do you mean by functional analysis being applied combinatorics.
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Erik DavisApr 15 '10 at 1:01

I never understood this quote, I mean, there's nasty algebra and nice algebra, so isn't that statement a bit strong?
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Harry GindiNov 30 '09 at 6:28

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I actually think it refers to something Mephistophelian about algebra, as if such heights of abstraction are meant not for mortals, or such symbolic calculation lacks intuitive "soul". (That would be a tendentious way of putting it!)
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Todd Trimble♦Jan 31 '11 at 1:39

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Oh, and I swear that I wrote that before seeing Anton Petrunin's contribution!
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Todd Trimble♦Jan 31 '11 at 1:42

"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate.

"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." - Grothendieck

" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?" "- Serge Lang

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

--Stefan Banach

"Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies."